Singular integral

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator

${\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,}$

whose kernel function K : Rⁿ×Rⁿ → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|⁻ⁿ asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on L^p(Rⁿ).